Sine and cosine?

Question

Can someone just tell me how they are linked together?
And what the relationship between the functions y = sin x and y = cos x?
just really simply....

Answer 1

y = cos x is y = sin x with a phase shift of 90 degrees

sine wave starts at zero, goes up to one, back to zero, to negative one starts all over

cosine wave starts at one, down to zero, down to negative one, up to zero again, starts all over

Answer 2

The easy way to visualize sines and cosines is to consider them as ratios of the sides of a right triangle. That is, for a given angle (other than the 90 degree one), the opposite side over the hypotenuse is equal to the sine, and the adjacent side over the hypotenuse is equal to the cosine. The relationship between the two then comes from the Pythagorean theorem, where the the sum of the sides squared equals the square of the hypotenuse. If you write that in terms of the sines and cosines, you result in the equation that the (sin x)**2 + (cos x)**2 = 1. (where ** means "squared"). Incidentally, if you're dealing with a right triangle, than sin x and cos y have the same value, where x and y are the two non 90 degree angles. Good luck.

Answer 3

Most important, (sin x)^2 + (cos x)^2 = 1

Also, cos x = sin(pi/2 - x), which is why the cosine is called the cosine: it's the sine of the complementary angle.

Answer 4

Draw a quadrant of a circle between the x and y axes.
Pick any point (x,y) on the arc and draw a radius from the origin to the point. Call the angle between the radius θ. Now draw lines from your point perpendicular to the x and y axes, and you will have a rectangle.
The definitions are:
sinθ = y/r (side opposite)/(hypotenuse)
cosθ = x/r (side adjacent)/hypotenuse)
Now observe that the angle between your radius and the y-axis is (90° - θ). Now,
sin(90° - θ) = x/r = cosθ
cos(90° - θ) = y/r = sinθ

Another relation exists between sin and cos:
Note that x^2 + y^2 = r^2
then
r^2cos^2θ + r^2 sin^2θ = r^2
Dividing through by r,
cos^2θ + sin^2θ = 1

Answer 5

in radians, the cosine graph is shifted pi/2 to the left of the sine graph, in degrees, 90

Answer 6

sin(x) = cos(x - (pi/2)) or cos(x - 90)
cos(x) = sin(x + (pi/2)) or sin(x + 90)

cos(45) = sin(45)
cos(0) = sin(90)
cos(180) = sin(270)
cos(270) = sin(0) or sin(180)